Phase-space consistency of stellar dynamical models determined by separable augmented densities

Assuming the separable augmented density, it is always possible to construct a distribution function of a spherical population with any given density and anisotropy. We consider under what conditions the distribution constructed as such is in fact non-negative everywhere in the accessible phase space. We first generalize the known necessary conditions on the augmented density using fractional calculus. The condition on the radius part R(r(2)) (whose logarithmic derivative is the anisotropy parameter) is equivalent to the complete monotonicity of omega R-1(omega(-1)). The condition on the potential part on the other hand is given by its derivative up to any order not greater than 3/2 - beta(0) being non-negative where beta(0) is the central anisotropy parameter. We also derive a specialized inversion formula for the distribution from the separable augmented density, which leads to sufficient conditions on separable augmented densities for the non-negativity of the distribution. These last conditions are generalizations of the similar condition derived earlier for the generalized Cuddeford system to arbitrary separable systems.